A117431 String n is at position n in decimal digits of golden ratio (phi).

1, 20, 62, 9956

Offset: 1

Author

Colin Rose, Mar 14 2006

Keywords

• nonn
• base
• hard
• more

Comments

The next such number is greater than 10^7. Not only does number 20 occur at the 20th digit, but it occurs again as the 20th pair of digits (cf. A117432).

Examples

			1 is a term because the first digit in the golden ratio phi is 1. (phi = 1.6180339887498948482045 ...)
		

Links

• Eric Weisstein's World of Mathematics, The Golden Ratio

Crossrefs

Cf. A001622, A057679, A117432.

Programs

Mathematica

StringFinder[m_] := Module[{cc = 10^m + m, sol, aa}, sol = Partition[RealDigits[(1+Sqrt[5])/2, 10, cc] // First, m, 1]; Do[aa = FromDigits[sol[[i]]]; If[aa==i, Print[{i, aa}]], {i,Length[sol]}];] (* Example: StringFinder[2] produces all 2-digit members of the sequence. *)

Python

from sympy import S
def aupto(nn):
  phistr = str(S.GoldenRatio.n(nn+len(str(nn))+1)).replace(".", "")[:-1]
  for n in range(1, nn+1):
    nstr = str(n)
    if phistr[n-1:n-1+len(nstr)] == nstr: print(n, end=", ")
aupto(10**5) # Michael S. Branicky, Jan 20 2021